In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV. We adopt the following convention: for all $x = (x_1, \ldots, x_n) \in \Lambda_n$, we denote $x_{n+1} = x_1$ and $x_0 = x_n$.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Show that $Z_n(h) = \operatorname{tr}(A^n)$, where $\operatorname{tr}$ denotes the trace of a square matrix.

We assume in this part that all roots of $p$ are stable and have multiplicity 1 and we denote by $h = Xp'$ (where $p'$ is the derivative polynomial of $p$) and $h_0$ the reciprocal polynomial of $h$. We recall that, according to question 3, there exists a real number $\lambda \in \{-1, 1\}$ such that $p = \lambda p_0$.
Show that there exists a real number $\eta > 0$ such that for every $r \in ]1-\eta; 1[$, the polynomial $p(rX)$ is split, has exactly $\sigma(p)$ roots inside the interval $]-1; 1[$ and has no stable root.

In this subsection, we assume that $J_n = J_n^{(\mathrm{C})}$, the matrix introduced in subsection A-II.
For all $x \in \Lambda_n$, we set $s_n(x) = \sum_{k=1}^n x_i$. Verify that $$Z_n(h) = \mathrm{e}^{-\frac{\beta}{2}} \sum_{x \in \Lambda_n} \exp\left(\frac{\beta}{2n}\left(s_n(x)\right)^2 + h s_n(x)\right).$$

Let $n \geq 1$ be an integer. We say that a matrix $M \in \mathcal{M}_n(\mathbb{R})$ is doubly stochastic if for all $i, j \in \{1, \ldots, n\}$ we have $$M_{ij} \geq 0 \quad \text{and} \quad \sum_{k=1}^n M_{ik} = \sum_{k=1}^n M_{kj} = 1.$$ We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$.
Show that $B_n$ is a polytope and determine its dimension.

Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $S_n$ the symmetric group of order $n$. For all $\sigma \in S_n$, we define $P^\sigma \in \mathcal{M}_n(\mathbb{R})$ as follows: for $i, j \in \{1, 2, \ldots, n\}$ we set $P^\sigma_{ij} = 1$ if $j = \sigma(i)$, $P^\sigma_{ij} = 0$ otherwise. Show that $P^\sigma$ is a vertex of $B_n$ for all $\sigma \in S_n$.

Let $n \geq 1$ be an integer. We denote by $B_n$ the set of doubly stochastic matrices in $\mathcal{M}_n(\mathbb{R})$ and $\mathcal{M}_n(\mathbb{Z})$ the set of $n \times n$ matrices with integer coefficients.
Suppose that $M \in B_n \backslash \mathcal{M}_n(\mathbb{Z})$. Show that there exists a sequence $(r_1, s_1), (r_2, s_2), \ldots, (r_k, s_k)$ of pairs of indices with $k \geq 2$ such that $$0 < M_{r_i, s_i} < 1, \quad 0 < M_{r_i, s_{i+1}} < 1 \quad \text{and} \quad (r_k, s_k) = (r_1, s_1)$$ then that we can assume that all the pairs $(r_1, s_1), (r_1, s_2), (r_2, s_2), \ldots, (r_{k-1}, s_{k-1}), (r_{k-1}, s_k)$ are distinct.

139- From the matrix relation $\begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix} A \begin{bmatrix} 5 & 2 \\ 3 & 1 \end{bmatrix} = \begin{bmatrix} 3 & 0 \\ -1 & 2 \end{bmatrix}$, the first row of matrix $A$ is which of the following?
(1) $\begin{bmatrix} 12 & -17 \end{bmatrix}$ (2) $\begin{bmatrix} -21 & 30 \end{bmatrix}$ (3) $\begin{bmatrix} -17 & 30 \end{bmatrix}$ (4) $\begin{bmatrix} 12 & -21 \end{bmatrix}$

140- If $A = \begin{bmatrix} 0 & -\tan\alpha \\ \tan\alpha & 0 \end{bmatrix}$ and $I$ is the identity matrix of order 2, the first row of $(I+A)(I-A)^{-1}$ is which of the following?
(1) $[\cos 2\alpha \;\; -\sin 2\alpha]$ (2) $[\cos 2\alpha \;\; \sin 2\alpha]$ (3) $[\sin 2\alpha \;\; \cos 2\alpha]$ (4) $[-\sin 2\alpha \;\; \cos 2\alpha]$

139- If matrix $A$ has the transformation $T(x,y) = (2x - y,\ 3x - 4y)$ and $I$ is the identity matrix, and $\alpha$ and $\beta$ are two real numbers such that $\alpha A + \beta I = A^{-1}$, what is the value of $\beta$?
(1) $-\dfrac{3}{5}$ (2) $-\dfrac{1}{5}$ (3) $\dfrac{2}{5}$ (4) $\dfrac{4}{5}$

139- If $A = [a_{ij}]_{r \times 3}$ and $B = [b_{ij}]_{r \times 2}$, which of the following matrix products is defined?
(1) $AB$ (2) $A^t B$ (3) $B^t A^t$ (4) $AB^t$

140- If $A = \begin{bmatrix} 1 & -1 & 2 \\ 0 & 2 & -1 \\ 0 & 0 & 3 \end{bmatrix}$, what is the sum of the entries of the second column of $A^{-1}$?
(1) $-\dfrac{1}{3}$ (2) $\dfrac{2}{3}$
(3) $1$ (4) zero

139- If two matrices $A$ and $(I - A)$ are inverses of each other, then matrix $A^4$ equals which of the following?
(1) $A$ (2) $-A$ (3) $I$ (4) $-I$

137. Matrix $A = \begin{bmatrix} 5 & 2 & -1 \\ 4 & 3 & -2 \\ 1 & 6 & 7 \end{bmatrix}$ is written as the sum of a symmetric matrix and a skew-symmetric matrix. The determinant of the symmetric matrix is:
(1) $16$ (2) $18$ (3) $22$ (4) $24$

139. If $\begin{bmatrix} \cos 15° & \sin 15° \\ -\sin 15° & \cos 15° \end{bmatrix}^n = -I$, what is the smallest natural number $n$?
(1) $6$ (2) $12$ (3) $18$ (4) $24$

138- If $A = \begin{bmatrix} 1 & 3 & 6 & 24 \\ \frac{1}{2} & 1 & 2 & 8 \end{bmatrix}$, $B = \begin{bmatrix} \frac{1}{6} & \frac{1}{2} & 1 & 4 \\ \frac{1}{24} & \frac{1}{8} & \frac{1}{4} & 1 \end{bmatrix}$, and $C = \begin{bmatrix} A \\ B \end{bmatrix}$, the sum of the main diagonal entries of matrix $C^T$ is which of the following?
(1) $16$ (2) $18$ (3) $20$ (4) $24$

139- The values of $x$ from the relation $\begin{vmatrix} 0 & x-3 & x-2 \\ x+3 & 0 & -4 \\ x+2 & 6 & 0 \end{vmatrix} = 0$ are which of the following?
(1) $-1, -6$ (2) $-1, 6$ (3) $1, -6$ (4) $1, 6$
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140-- Three pages with matrix equations. If $\begin{vmatrix} a & -1 & 3 \\ b & 2 & 4 \\ c & -2 & 1 \end{vmatrix} = 5$, then the three pages intersect. With which of the following lengths do they intersect?
$$\begin{bmatrix} 2 & -1 & 3 \\ 1 & 2 & 4 \\ 3 & -2 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} a \\ b \\ c \end{bmatrix}$$
(1) $-\dfrac{1}{3}$ (2) $-\dfrac{1}{2}$ (3) $\dfrac{1}{3}$ (4) $\dfrac{1}{2}$

130- From the matrix relation $\begin{bmatrix} 3 & -1 & 1 \\ 4 & 0 & -2 \\ 1 & 2 & 0 \end{bmatrix} \begin{bmatrix} x \\ 2x \\ -1 \end{bmatrix} = 0$, $\begin{bmatrix} x & 2x & -1 \end{bmatrix}$, the nonzero value of $x$ is which of the following?
(1) $\dfrac{2}{9}$ (2) $\dfrac{3}{8}$ (3) $\dfrac{4}{9}$ (4) $\dfrac{3}{5}$

131- If $A = \begin{bmatrix} 2 & 1 \\ 4 & 2 \end{bmatrix}$, from the relation $AX = A - 2I$, matrix $X$ is which of the following?
(1) $\begin{bmatrix} -2 & 2 \\ 3 & -1 \end{bmatrix}$ (2) $\begin{bmatrix} -2 & 1 \\ 4 & -1 \end{bmatrix}$ (3) $\begin{bmatrix} 2 & -1 \\ 4 & 2 \end{bmatrix}$ (4) $\begin{bmatrix} 1 & 2 \\ 4 & -1 \end{bmatrix}$

132- If $A$ is a $3 \times 3$ matrix and $|A| = 4$, then the determinant of matrix $A \cdot A$ is which of the following?
(1) $64$ (2) $96$ (3) $128$ (4) $256$

138- If $A = \begin{bmatrix} x & -1 & -x \\ 0 & 0 & 4 \\ y & z & z \end{bmatrix}$, $B = \begin{bmatrix} yz & \frac{1}{2} & 2 \\ yz & 0 & -4y \\ 0 & \frac{1}{2} & 0 \end{bmatrix}$ and matrix $AB$ is scalar for every $y \in \mathbb{Z}$, the value of $xy$ is which?
\[ \text{(1)}\ -1 \qquad \text{(2)}\ -2 \qquad \text{(3)}\ 1 \qquad \text{(4)}\ 2 \]

139- If $A = \begin{bmatrix} 1 & -1 & -3 \\ 4 & 1 & 2 \\ 2 & 1 & 3 \end{bmatrix}$ and matrix $X$ satisfies the matrix equation $X = \begin{bmatrix} 3 & 0 \\ -2 & 1 \end{bmatrix}$, $$\begin{bmatrix} 2|A| & |A| \\ 1 & \dfrac{2}{|A|} \end{bmatrix} X = \begin{bmatrix} 3 & 0 \\ -2 & 1 \end{bmatrix}$$ holds. The smallest main diagonal entry of matrix $X$ is which?
\[ \text{(1)}\ -15 \qquad \text{(2)}\ -3 \qquad \text{(3)}\ 6 \qquad \text{(4)}\ 8 \]

33. If $A = \begin{bmatrix} \log_5^2 & \log_5^2 \\ \log_5^2 & \log_5^2 \end{bmatrix}$ and $B = \begin{bmatrix} 6|A| & 2|A| \\ 3|A| & 36|A| \end{bmatrix}$, what is the determinant of $B$?

• [(1)] $\dfrac{9}{4}$
• [(2)] $\dfrac{15}{4}$
• [(3)] $\dfrac{9}{8}$
• [(4)] $\dfrac{15}{8}$

A ``basic row operation'' on a matrix means adding a multiple of one row to another row. Consider the matrices $$A = \left(\begin{array}{rrr} x & 5 & x \\ 1 & 3 & -2 \\ -2 & -2 & 2 \end{array}\right) \quad \text{and} \quad B = \left(\begin{array}{rrr} 0 & 0 & 21 \\ 1 & -1 & -14 \\ 0 & \frac{4}{3} & 4 \end{array}\right)$$ It is given that $B$ can be obtained from $A$ by applying finitely many basic row operations. Then, the value of $x$ is:
(A) 3
(B) $-3$
(C) $-1$
(D) 2.

Let $\theta = \frac{2\pi}{7}$ and consider the following matrix $$A = \left(\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right)$$ If $A^n$ means $A \times \cdots \times A$ ($n$ times), then $A^{100}$ is
(A) $\left(\begin{array}{rr} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{array}\right)$
(B) $\left(\begin{array}{rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right)$
(C) $\left(\begin{array}{ll} 1 & 0 \\ 0 & 1 \end{array}\right)$
(D) $\left(\begin{array}{rr} 0 & -1 \\ 1 & 0 \end{array}\right)$.